The Grassmann-like Manifold of Centered Planes
- 7 Downloads
Connections associated with the Grassmann-like manifold of centered planes in the multidimensional projective space are studied. A geometric interpretation of these connections in terms of maps and translations of equipping planes is given. An intrinsic analog of Norden’s strong normalization of the manifold under consideration is constructed.
KeywordsCartan’s exterior form method Grassmann manifold Norden’s normalization average connection
Unable to display preview. Download preview PDF.
- 1.S. P. Finikov, Cartan’s Method of Exterior Forms inDifferentialGeometry. The Theory of Compatibility of Systems of Total and Partial Differential Equations (GITTL, Moscow–Leningrad, 1948) [in Russian].Google Scholar
- 4.Yu. I. Shevchenko, “Laptev’s and Lumiste’s tricks for specifying a connection in a principal bundle,” Differ. Geom.Mnogoobr. Figur 37, 179–187 (2006).Google Scholar
- 5.A. A. Borisenko and Yu. A. Nikolaevskii, “Grassmannmanifolds and theGrassmann image of submanifolds,” UspekhiMat. Nauk 46 (2 (278)), 41–83 (1991) [RussianMath. Surveys 46 (2), 45–94 (1991)].Google Scholar
- 6.Yu. G. Lumiste, “Induced connections in immersed projective and affine bundles,” in Works inMathematics and Mechanics, Vol. 177, Uchenye Zapiski Tartuskogo Universiteta (Tartuskii Gos. Univ., Tartu, 1965), pp. 6–41 [in Russian].Google Scholar
- 11.D. Baralic, “How to understand Grassmannians?,” The Teaching of Mathematics 14 (2), 147–157 (2011).Google Scholar
- 12.M. M. Postnikov, Lectures in Geometry, Semester II: Linear Algebra and Differential Geometry (Nauka, Moscow, 1979;Mir, Moscow, 1983).Google Scholar
- 13.G. F. Laptev, “Differential geometry of immersed manifolds. Group-theoreticmethod of differential-geometric research,” in Trudy Moskov. Mat. Obshch. (GITTL, Moscow, 1953), Vol. 2, pp. 275–382 [in Russian].Google Scholar
- 14.Yu. I. Shevchenko, Equipments of Central-Projective Manifolds (Kaliningrad. Gos. Univ., Kaliningrad, 2000) [in Russian].Google Scholar
- 17.B. N. Shapukov, Problems on Lie Groups and Their Applications (Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2002) [in Russian].Google Scholar
- 18.Yu. I. Shevchenko, “Parallel translations on a surface,” Differ. Geom. Mnogoobr. Figur 10, 154–158 (1979).Google Scholar
- 20.A. V. Chakmazyan, “A connection in normal bundles of normalized submanifolds V m in P n,” in Problems in Geometry, Vol. 10, Itogi Nauki i Tekhniki (VINITI, Moscow, 1978), pp. 55–74 [J. Soviet Math. 14 (3), 1205–1216 (1980)].Google Scholar